An article in the Journal of the Environmental Engineering Division (“Least Squares Estimates of BOD Parameters,” Vol. 106, 1980, pp. 1197–1202) describes the results of testing a sample from the Holston River below Kingport, TN, during August 1977. The biochemical oxygen demand (BOD) test is conducted over a period of time in days.

The resulting data is shown here:

Time (days): 1 2 4 6 8 10 12 14 16 18 20 BOD (mg/liter): 0.6 0.7 1.5 1.9 2.1 2.6 2.9 3.7 3.5 3.7 3.8

(a) Assuming that a simple linear regression model is appropriate, fit the regression model relating BOD ( y) to the time (x). What is the estimate of

(b) What is the estimate of expected BOD level when the time is 15 days?

(c) What change in mean BOD is expected when the time increases by 3 days?

(d) Suppose the time used is 6 days. Calculate the fitted value of y and the corresponding residual.

(e) Calculate the fitted for each value of used to fit the model. Then construct a graph of versus the corresponding observed values and comment on what this plot would look like if the relationship between y and x was a deterministic (no random error) straight line. Does the plot actually obtained indicate that time is an effective regressor variable in predicting BOD?